Regularized proportional odds models

The proportional odds model (POM) is commonly used in regression analysis to predict the outcome for an ordinal response variable. The maximum likelihood estimation (MLE) approach is typically used to obtain the parameter estimates. The likelihood estimates do not exist when the number of parameters, p, is greater than the number of observations n. The MLE also does not exist if there are no overlapping observations in the data. In a situation where the number of parameters is less than the sample size but p is approaching to n, the likelihood estimates may not exist, and if they exist they may have quite large standard errors. An estimation method is proposed to address the last two issues, i.e. complete separation and the case when p approaches n, but not the case when p>n. The proposed method does not use any penalty term but uses pseudo-observations to regularize the observed responses by downgrading their effect so that they become close to the underlying probabilities. The estimates can be computed easily with all commonly used statistical packages supporting the fitting of POMs with weights. Estimates are compared with MLE in a simulation study and an application to the real data.     

Dealing with separation or near-to-separation in the model for multinomial response with application to childhood health seeking behavior data from a complex survey

Separation or monotone-likelihood can be observed in fitting process of a multinomial logistic model using maximum likelihood estimation (MLE) when sample size is small and/or one of the outcome categories is rare and/or there is one or more influential covariates, resulting in infinite or biased estimate of at least one regression coefficient of the model. This study investigated empirically to identify the optimal data condition to define both ‘separation’ and ‘near-to-separation’ (partial separation) and explored their consequences in MLE and provided a solution by applying a penalized likelihood approach, which has been proposed in the literature, by adding a Jeffreys prior-based penalty term to the original likelihood function to remove the first-order bias in the MLEs of the multinomial logit model via equivalent Poisson regression. Furthermore, the penalized estimating equation (PMLE) is extended to a weighted estimating equation allowing for survey-weight for analyzing data from a complex survey. The simulation study suggests that the PMLE outperforms the MLE by providing smaller amount of bias and mean squared of error and better coverage. The methods are applied to analyze data on choice of health facility for treatment of childhood diseases.     

Kernel-based semiparametric multinomial logit modelling of political party preferences

Conventional, parametric multinomial logit models are in general not sufficient for capturing the complex structures of electorates. In this paper, we use a semiparametric multinomial logit model to give an analysis of party preferences along individuals’ characteristics using a sample of the German electorate in 2006. Germany is a particularly strong case for more flexible nonparametric approaches in this context, since due to the reunification and the preceding different political histories the composition of the electorate is very complex and nuanced. Our analysis reveals strong interactions of the covariates age and income, and highly nonlinear shapes of the factor impacts for each party’s likelihood to be supported. Notably, we develop and provide a smoothed likelihood estimator for semiparametric multinomial logit models, which can be applied also in other application fields, such as, e.g., marketing.     

Fixed Effects and Bias Due to the Incidental Parameters Problem in the Tobit Model

The maximum likelihood estimator (MLE) in nonlinear panel data models with fixed effects is widely understood (with a few exceptions) to be biased and inconsistent when T, the length of the panel, is small and fixed. However, there is surprisingly little theoretical or empirical evidence on the behavior of the estimator on which to base this conclusion. The received studies have focused almost exclusively on coefficient estimation in two binary choice models, the probit and logit models. In this note, we use Monte Carlo methods to examine the behavior of the MLE of the fixed effects tobit model. We find that the estimator's behavior is quite unlike that of the estimators of the binary choice models. Among our findings are that the location coefficients in the tobit model, unlike those in the probit and logit models, are unaffected by the “incidental parameters problem.” But, a surprising result related to the disturbance variance emerges instead - the finite sample bias appears here rather than in the slopes. This has implications for estimation of marginal effects and asymptotic standard errors, which are also examined in this paper. The effects are also examined for the probit and truncated regression models, extending the range of received results in the first of these beyond the widely cited biases in the coefficient estimators.     

Data augmentation,frequentist estimation,and the Bayesian analysis of multinomial logit models

This article describes a convenient method of selecting Metropolis– Hastings proposal distributions for multinomial logit models. There are two key ideas involved. The first is that multinomial logit models have a latent variable representation similar to that exploited by Albert and Chib (J Am Stat Assoc 88:669–679, 1993) for probit regression. Augmenting the latent variables replaces the multinomial logit likelihood function with the complete data likelihood for a linear model with extreme value errors. While no conjugate prior is available for this model, a least squares estimate of the parameters is easily obtained. The asymptotic sampling distribution of the least squares estimate is Gaussian with known variance. The second key idea in this paper is to generate a Metropolis–Hastings proposal distribution by conditioning on the estimator instead of the full data set. The resulting sampler has many of the benefits of so-called tailored or approximation Metropolis–Hastings samplers. However, because the proposal distributions are available in closed form they can be implemented without numerical methods for exploring the posterior distribution. The algorithm converges geometrically ergodically, its computational burden is minor, and it requires minimal user input. Improvements to the sampler’s mixing rate are investigated. The algorithm is also applied to partial credit models describing ordinal item response data from the 1998 National Assessment of Educational Progress. Its application to hierarchical models and Poisson regression are briefly discussed.     

A self-consistency approach to multinomial logit model with random effects

The computation in the multinomial logit mixed effects model is costly especially when the response variable has a large number of categories, since it involves high-dimensional integration and maximization. Tsodikov and Chefo (2008) developed a stable MLE approach to problems with independent observations, based on generalized self-consistency and quasi-EM algorithm developed in Tsodikov (2003). In this paper, we apply the idea to clustered multinomial response to simplify the maximization step. The method transforms the complex multinomial likelihood to Poisson-type likelihood and hence allows for the estimates to be obtained iteratively solving a set of independent low-dimensional problems. The methodology is applied to real data and studied by simulations. While maximization is simplified, numerical integration remains the dominant challenge to computational efficiency.     

AN ALTERNATIVE PARAMETRIC APPROACH FOR DISCRETE MISSING DATA PROBLEMS

We propose an iterative method of estimation for discrete missing data problems that is conceptually different from the Expectation–Maximization (EM) algorithm and that does not in general yield the observed data maximum likelihood estimate (MLE). The proposed approach is based conceptually upon weighting the set of possible complete-data MLEs. Its implementation avoids the expectation step of EM, which can sometimes be problematic. In the simple case of Bernoulli trials missing completely at random, the iterations of the proposed algorithm are equivalent to the EM iterations. For a familiar genetics-oriented multinomial problem with missing count data and for the motivating example with epidemiologic applications that involves a mixture of a left censored normal distribution with a point mass at zero, we investigate the finite sample performance of the proposed estimator and find it to be competitive with that of the MLE. We give some intuitive justification for the method, and we explore an interesting connection between our algorithm and multiple imputation in order to suggest an approach for estimating standard errors.     

Maximum likelihood estimation for incomplete multinomial data via the weaver algorithm

In a multinomial model, the sample space is partitioned into a disjoint union of cells. The partition is usually immutable during sampling of the cell counts. In this paper, we extend the multinomial model to the incomplete multinomial model by relaxing the constant partition assumption to allow the cells to be variable and the counts collected from non-disjoint cells to be modeled in an integrated manner for inference on the common underlying probability. The incomplete multinomial likelihood is parameterized by the complete-cell probabilities from the most refined partition. Its sufficient statistics include the variable-cell formation observed as an indicator matrix and all cell counts. With externally imposed structures on the cell formation process, it reduces to special models including the Bradley–Terry model, the Plackett–Luce model, etc. Since the conventional method, which solves for the zeros of the score functions, is unfruitful, we develop a new approach to establishing a simpler set of estimating equations to obtain the maximum likelihood estimate (MLE), which seeks the simultaneous maximization of all multiplicative components of the likelihood by fitting each component into an inequality. As a consequence, our estimation amounts to solving a system of the equality attainment conditions to the inequalities. The resultant MLE equations are simple and immediately invite a fixed-point iteration algorithm for solution, which is referred to as the weaver algorithm. The weaver algorithm is short and amenable to parallel implementation. We also derive the asymptotic covariance of the MLE, verify main results with simulations, and compare the weaver algorithm with an MM/EM algorithm based on fitting a Plackett–Luce model to a benchmark data set.     

Estimation of the scale parameter of the half-logistic distribution with multiply type II censored sample

In this paper, we consider the maximum likelihood and Bayes estimation of the scale parameter of the half-logistic distribution based on a multiply type II censored sample. However, the maximum likelihood estimator(MLE) and Bayes estimator do not exist in an explicit form for the scale parameter. We consider a simple method of deriving an explicit estimator by approximating the likelihood function and discuss the asymptotic variances of MLE and approximate MLE. Also, an approximation based on the Laplace approximation (Tierney & Kadane, 1986) is used to obtain the Bayes estimator. In order to compare the MLE, approximate MLE and Bayes estimates of the scale parameter, Monte Carlo simulation is used.     

Pooling choices or categories in multinomial logit models

The estimation of multinomial logit models today is routine. With this increased use has also come a need for testing. A test to determine whether choices can be combined is important. This paper presents a likelihood ratio test for combining choices in multinomial logit models. The use of the test is demonstrated with a simple example.     

Inadmissibility results of the mle for the multinomial problem when the parameter space is restricted or truncated

In binomial or multinomial problems when the parameter space is restricted or truncated to a subset of the natural parameter space, the maximum likelihood estimator (MLE) may be inadmissible under squared error loss. A quite general condition for the inadmissibility of MLEs in such cases can be established using the stepwise Bayes technique and the complete class theorem of Brown.     

A simple root selection method for univariate finite normal mixture models

It is well known that there exist multiple roots of the likelihood equations for finite normal mixture models. Selecting a consistent root for finite normal mixture models has long been a challenging problem. Simply using the root with the largest likelihood will not work because of the spurious roots. In addition, the likelihood of normal mixture models with unequal variance is unbounded and thus its maximum likelihood estimate (MLE) is not well defined. In this paper, we propose a simple root selection method for univariate normal mixture models by incorporating the idea of goodness of fit test. Our new method inherits both the consistency properties of distance estimators and the efficiency of the MLE. The new method is simple to use and its computation can be easily done using existing R packages for mixture models. In addition, the proposed root selection method is very general and can be also applied to other univariate mixture models. We demonstrate the effectiveness of the proposed method and compare it with some other existing methods through simulation studies and a real data application.     

Two-parameter ridge estimator in the binary logistic regression

The binary logistic regression is a commonly used statistical method when the outcome variable is dichotomous or binary. The explanatory variables are correlated in some situations of the logit model. This problem is called multicollinearity. It is known that the variance of the maximum likelihood estimator (MLE) is inflated in the presence of multicollinearity. Therefore, in this study, we define a new two-parameter ridge estimator for the logistic regression model to decrease the variance and overcome multicollinearity problem. We compare the new estimator to the other well-known estimators by studying their mean squared error (MSE) properties. Moreover, a Monte Carlo simulation is designed to evaluate the performances of the estimators. Finally, a real data application is illustrated to show the applicability of the new method. According to the results of the simulation and real application, the new estimator outperforms the other estimators for all of the situations considered.     

Regression Analysis of Multivariate Fractional Data

The present article discusses alternative regression models and estimation methods for dealing with multivariate fractional response variables. Both conditional mean models, estimable by quasi-maximum likelihood, and fully parametric models (Dirichlet and Dirichlet-multinomial), estimable by maximum likelihood, are considered. A new parameterization is proposed for the parametric models, which accommodates the most common specifications for the conditional mean (e.g., multinomial logit, nested logit, random parameters logit, dogit). The text also discusses at some length the specification analysis of fractional regression models, proposing several tests that can be performed through artificial regressions. Finally, an extensive Monte Carlo study evaluates the finite sample properties of most of the estimators and tests considered.     

A gradient-based algorithm for semiparametric models with missing covariates

In the parametric regression model, the covariate missing problem under missing at random is considered. It is often desirable to use flexible parametric or semiparametric models for the covariate distribution, which can reduce a potential misspecification problem. Recently, a completely nonparametric approach was developed by [H.Y. Chen, Nonparametric and semiparametric models for missing covariates in parameter regression, J. Amer. Statist. Assoc. 99 (2004), pp. 1176–1189; Z. Zhang and H.E. Rockette, On maximum likelihood estimation in parametric regression with missing covariates, J. Statist. Plann. Inference 47 (2005), pp. 206–223]. Although it does not require a model for the covariate distribution or the missing data mechanism, the proposed method assumes that the covariate distribution is supported only by observed values. Consequently, their estimator is a restricted maximum likelihood estimator (MLE) rather than the global MLE. In this article, we show the restricted semiparametric MLE could be very misleading in some cases. We discuss why this problem occurs and suggest an algorithm to obtain the global MLE. Then, we assess the performance of the proposed method via some simulation experiments.     

Bias-correction for Weibull common shape estimation

A general method for correcting the bias of the maximum likelihood estimator (MLE) of the common shape parameter of Weibull populations, allowing a general right censorship, is proposed in this paper. Extensive simulation results show that the new method is very effective in correcting the bias of the MLE, regardless of censoring mechanism, sample size, censoring proportion and number of populations involved. The method can be extended to more complicated Weibull models.     

Small area estimates of labour force participation under a multinomial logit mixed model

Summary.  A new methodology is developed for estimating unemployment or employment characteristics in small areas, based on the assumption that the sample totals of unemployed and employed individuals follow a multinomial logit model with random area effects. The method is illustrated with UK labour force data aggregated by sex–age groups. For these data, the accuracy of direct estimates is poor in comparison with estimates that are derived from the multinomial logit model. Furthermore, two different estimators of the mean-squared errors are given: an analytical approximation obtained by Taylor linearization and an estimator based on bootstrapping. A simulation study for comparison of the two estimators shows the good performance of the bootstrap estimator.     

Robust estimation of the mean vector for high-dimensional data set using robust clustering

The first step in statistical analysis is the parameter estimation. In multivariate analysis, one of the parameters of interest to be estimated is the mean vector. In multivariate statistical analysis, it is usually assumed that the data come from a multivariate normal distribution. In this situation, the maximum likelihood estimator (MLE), that is, the sample mean vector, is the best estimator. However, when outliers exist in the data, the use of sample mean vector will result in poor estimation. So, other estimators which are robust to the existence of outliers should be used. The most popular robust multivariate estimator for estimating the mean vector is S-estimator with desirable properties. However, computing this estimator requires the use of a robust estimate of mean vector as a starting point. Usually minimum volume ellipsoid (MVE) is used as a starting point in computing S-estimator. For high-dimensional data computing, the MVE takes too much time. In some cases, this time is so large that the existing computers cannot perform the computation. In addition to the computation time, for high-dimensional data set the MVE method is not precise. In this paper, a robust starting point for S-estimator based on robust clustering is proposed which could be used for estimating the mean vector of the high-dimensional data. The performance of the proposed estimator in the presence of outliers is studied and the results indicate that the proposed estimator performs precisely and much better than some of the existing robust estimators for high-dimensional data.     

Robust and efficient estimation of GARCH models based on Hellinger distance

It is well known that financial data frequently contain outlying observations. Almost all methods and techniques used to estimate GARCH models are likelihood-based and thus generally non-robust against outliers. Minimum distance method, as an important tool for statistical inferences and a competitive alternative for achieving robustness, has surprisingly not been well explored for GARCH models. In this paper, we proposed a minimum Hellinger distance estimator (MHDE) and a minimum profile Hellinger distance estimator (MPHDE), depending on whether the innovation distribution is specified or not, for estimating the parameters in GARCH models. The construction and investigation of the two estimators are quite involved due to the non-i.i.d. nature of data. We proved that the MHDE is a consistent estimator and derived its bias in explicit expression. For both of the proposed estimators, we demonstrated their finite-sample performance through simulation studies and compared with the well-established methods including MLE, Gaussian Quasi-MLE, Non-Gaussian Quasi-MLE and Least Absolute Deviation estimator. Our numerical results showed that MHDE and MPHDE have much better performance than MLE-based methods when data are contaminated while simultaneously they are very competitive when data is clean, which testified to the robustness and efficiency of the two proposed MHD-type estimations.     

Estimation of the scale parameter of the half-logistic distribution under progressively type II censored sample

Based on a progressively type II censored sample, the maximum likelihood and Bayes estimators of the scale parameter of the half-logistic distribution are derived. However, since the maximum likelihood estimator (MLE) and Bayes estimator do not exist in an explicit form for the scale parameter, we consider a simple method of deriving an explicit estimator by approximating the likelihood function and derive the asymptotic variances of MLE and approximate MLE. Also, an approximation based on the Laplace approximation (Tierney and Kadane in J Am Stat Assoc 81:82–86, 1986) and importance sampling methods are used for obtaining the Bayes estimator. In order to compare the performance of the MLE, approximate MLE and Bayes estimates of the scale parameter, we use Monte Carlo simulation.